Find the sum of the three integers

Let the the three positive integers be $x,x+1$ and $x+2$ . From the data given, we have

$x^2+(x+1)^2+(x+2)^2=29$

$x^2+x^2+2x+1+x^2+4x+4=29$

$3x^2+6x-24=0$

$x^2+2x-8=0$

By factoring, we get

$(x+4)(x-2)=0$

$x=-4$ and $x=2$

So the integers are $2,3$ and $4$ , and the sum $2+3+4=\boxed{9}$

Let the integers are (n – 1), n, (n + 1). The sum of the squares is 3n^2 + 2 = 29, n=3. So the sum of the three integers is 9.